An analysis of initial post-buckling of a micro-scaled beam with hinged-hinged ends based on the strain gradient elasticity is presented. At the first bifurcation point, the second variation of the potential energy is analytically proved to be positive semidefinite when the compression equals to the critical load. The fourth variation of the potential energy is positive when the disturbance of the lateral deflection matches the buckling mode. Based on Koiter initial post-buckling stability theory, the neutral equilibrium of straight state of the micro-scaled beam is stable at the stability limit. The bending shape in the initial post-buckling is stable when the compressive force exceeds slightly the critical load. Although the critical load of the micro-scale beam with hinged-hinged ends is influenced by size effects, the initial post-buckling behavior of it is similar to that of the classical beam.
K E Y W O R D Sbifurcation, micro-scaled, positive semidefinite, size effect, stability
In this paper, the initial post-buckling of a fixed–fixed strut in compression at the first bifurcation point is analyzed. Using the Fourier series of the lateral deflection, the second variation of the potential energy is proved, analytically, to be semi-positive definite when the compression is equal to the Euler critical load. The fourth variation of the potential energy is positive when the disturbance of the lateral deflection matches the buckling mode. Based on Koiter’s initial post-buckling theory, the equilibrium of the straight state of the strut is stable at the stability limit; when the compression slightly exceeds the Euler critical load, the curved shape at initial post-buckling is stable.
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